Question

The order and degree of the differential equation \( \sqrt{\frac{dy}{dx}} - 4 \frac{dy}{dx} - 7x = 0 \) are respectively

• A. \( 1 \, \text{and} \, \frac{1}{2} \)
• B. \( 2 \, \text{and} \, 1 \)
• C. \( -1 \, \text{and} \, 1 \)
• D. \( 1 \, \text{and} \, 2 \)

Hint:

To determine the degree of a differential equation, remove square roots or fractional powers. The order is identified by the highest derivative in the equation.

Answer 1

The Correct Option is D

The given differential equation is: \[ \sqrt{\frac{dy}{dx}} - 4 \frac{dy}{dx} - 7x = 0. \]

 Step 1: Simplify the equation. To identify the degree of the equation, first eliminate the square root by squaring both sides. Rearranging the terms gives: \[ \sqrt{\frac{dy}{dx}} = 4\frac{dy}{dx} + 7x. \] Squaring both sides results in: \[ \left( \sqrt{\frac{dy}{dx}} \right)^2 = \left( 4\frac{dy}{dx} + 7x \right)^2. \] This simplifies to: \[ \frac{dy}{dx} = 16\left(\frac{dy}{dx}\right)^2 + 49x^2 + 56\frac{dy}{dx}. \] 

Step 2: Determine the order and degree. 

The order of a differential equation is defined as the highest derivative present in the equation. Here, the highest derivative is \( \frac{dy}{dx} \), so the order is \( \mathbf{1} \). 
The degree refers to the power of the highest order derivative after removing any radicals or fractional powers. Upon squaring, the highest power of \( \frac{dy}{dx} \) is \( \mathbf{2} \).

 Final Answer: \[ \boxed{1 \, \text{and} \, 2} \]